<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:10"/><i><font color="Green">E11</font></i>: 
 <a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<b>assume </b><a NAME="E3:10"/><i><font color="Green">E12</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
(  <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1"/><i><font color="Green">E13</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1_1_1"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:10_1_1_1_1"/>
 <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E1:10_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/>
<a NAME="E4:10_1_1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#D19">INTEGRA1:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:10_1_1_1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> holds <br/> <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:10_1_1_1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:10_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E4:10_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:10_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:10_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E4:10_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E7:10_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E2:10_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<b>hence </b><a NAME="E8:10_1_1_1_1_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_1_1_1_1"/>
 <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E2:10_1_1_1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E3:10_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:10_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:10_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E4:10_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<a NAME="E5:10_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:10_1_1_1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:10_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E6:10_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="integra4.html#E4:10_1_1_1_1_2_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:10_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="integra4.html#E3:10_1_1_1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_1_1_1_1_2"/>
 <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E2:10_1_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:10_1_1_1_1"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T111">RVSUM_1:111</a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra4.html#E1:10_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="integra1.html#T26">INTEGRA1:26</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:10_1_1_1"/><b>then </b>
 <a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><a NAME="E4:10_1_1_1"/><b>then </b>
 <a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>3</sub></font> is <a href="fdiff_1.html#V1">convergent_to_0</a>
 <b>by </b><i><a class="txt" href="integra4.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="integra4.html#E3:10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_1"/>
(  <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="integra4.html#E1:10"><i><font color="Green">E10</font></i></a>, <a class="txt" href="integra4.html#E1:10_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="integra3.html#T19">INTEGRA3:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_2"/><i><font color="Green">E13</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:10_1_1_2"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:10_1_1_2_1"/>
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_2_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E2:10_1_1_2_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E1:10_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="integra4.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/>
<a NAME="E3:10_1_1_2_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:10_1_1_2_1_1"/><i><font color="Green">E17</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>5</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E18</font></i>:  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E2:10_1_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>
;<br/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>5</sub></font>,1 = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra4.html#E2:10_1_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra4.html#E2:10_1_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="integra4.html#E1:10_1_1_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E3:10_1_1_2_1_1_1"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>5</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:10_1_1_2_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E2:10_1_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E4:10_1_1_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="integra4.html#E1:10_1_1_2"><i><font color="Green">E13</font></i></a></i>
;<br/>
<a NAME="E7:10_1_1_2_1_1"/><b>then </b>
 <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:10_1_1_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E8:10_1_1_2_1_1"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_1_2_1_1"/>
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:10_1_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/><b>hence </b><a NAME="E4:10_1_1_2"/>
 <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/><a NAME="E5:10_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E2:10_1_1_2"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E6:10_1_1_2"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E3:10_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/><b>hence </b><a NAME="E7:10_1_1_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="integra4.html#E1:10_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="integra4.html#E7"><i><font color="Green">Lemma7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
(  <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
